What's your relationship to the child if you're having him for 4 days at a time, @Mumofmabes ? Grandparent? Co-parent? How often are you actually having him for 4 days?
If there's no diagnosis made, is his mum able to explain why she thinks he's autistic? Does he have social communication difficulties and features of autism across the domains? Do you have experience of autism which enables you to feel more confident in your own assessment than his mum's assessment? (If a parent says a child has a lifelong neurodevelopmental disorder which has significant implications for their future, they tend to have a reason for thinking that)
Complying with the instructions of a 'non-safe' person (by this I mean anyone around whom masking is necessary) is often a sensible thing that a 'high functioning' child can learn to do. Staying in a bedroom quietly when told to do so can, again, be a sensible thing to do, and actually may be an opportunity for an autistic child to hide away from the rest of a stressful environment in which they're having to mask. A child saying “I can be naughty and still get taken out" is not in itself proof of the child having control over their actions or of intentionally acting differently.
There's simply not enough here to say "Yes, this child's autistic" or "No they're not". All you can really take from the information you have is that for some reason the child behaves differently with you (note: not 'better' because while it might seem better for you, if the child is autistic then they're putting themselves through a lot of stress to achieve that and what might be 'better' for them would be a neuroaffirming approach so they don't have to mask for so long).
Rather than judging his mum and criticising her parenting and her judgement, it might be more constructive to suggest to her that the child get a proper assessment of their needs (she could speak with their school/education provider, GP or go privately) which could lead to a clearer picture for everyone of what's going on and how to support the child. It sounds like the child's mum is dealing with a lot, and some support could help but it would need to be the right support and advice. Eg techniques that work for neurotypical children often don't work for neurodiverse children. If she has a neurotypical child and is parenting as though they're autistic, that's not going to be very effective, and if she has an autistic child and parents as though they're neurotypical, that's not going to be effective either.